Hotel Kea
Hafnarstræti 87-89
Ran at around 16000 kr for a double and around 12500 kr for a single. the staff at the hotel is extremely helpful in explaining what to do and where to eat, even if you don't stay there!
There are many guesthouses more inland immediately away from the city center. They double as residences for college students, but they are the cheapest lodging you'll find, at around 8000 Kr double and 5000 single.
Gista (http://www.gista.is) is renting out apartments with Hotel standard. Located 3 min walk from the centre of Akureyri. Very good accomodation for groups, as the whole house 3 apartments can be rented, which can accomodate up to 16 people.